Question

If \[\left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right] = \left[ {25} \right]\] , where \[m,n \in N\] and \[m < n\] , then \[\left( {m,n} \right) = \]
A. \[\left( {2,3} \right)\]
B. \[\left( {3,4} \right)\]
C. \[\left( {4,3} \right)\]
D. \[\left( {3,2} \right)\]

Answer 1

Hint: In the above given problem, we are given two matrices of order \[1 \times 2\] and \[2 \times 1\] . Both the matrices contain two elements, they are the natural numbers \[m\] and \[n\] . The product of these two matrices is given as \[\left[ {25} \right]\] . We have to find the value of these two numbers \[m\] and \[n\] . In order to approach the solution, we have to use the method of multiplication of two matrices.

Complete step by step answer:
Given that, two matrices of order \[1 \times 2\] and \[2 \times 1\] respectively, which are \[\left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right]\] where \[m\] and \[n\] are natural numbers and \[m < n\] .
Their product is given as,
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right] = \left[ {25} \right]\]
We have to find the value of \[m\] and \[n\] .
Now consider only the product \[\left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right]\] ,

Using the method of multiplication of two matrices, we can write the product of above two matrices as,
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right] = \left[ {m \times m + n \times n} \right]\]
That can be written as,
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right] = \left[ {{m^2} + {n^2}} \right]\]

Now comparing the two products of \[\left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\] and \[\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right]\] , we have
\[ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  m&n
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  m \\
  n
\end{array}} \right] = \left[ {{m^2} + {n^2}} \right] = \left[ {25} \right]\]
Taking the products out of the matrices sign, that gives us the equation
\[ \Rightarrow {m^2} + {n^2} = 25\]
The only natural numbers that can satisfy the above condition are \[3\] and \[4\] , as we have
\[ \Rightarrow {3^2} + {4^2} = 9 + 16 = 25\]
But , here it is also given the condition that \[m < n\]. Therefore, we have \[m = 3\] and \[n = 4\]. Therefore, \[\left( {m,n} \right) = \left( {3,4} \right)\].

Hence, the correct option is B.

Note: When we do multiplication of two matrices, then we must keep in mind that the multiplication of two matrices is possible only when the number of columns of the 1st matrix is equal to the number of rows of the 2nd matrix. Hence, after the multiplication, the resulting product matrix will have the same number of rows as the 1st matrix, and the same number of columns as the 2nd matrix.